According to German outlet Bild (h/t Metro's Chris Davie), the Ligue 1 outfit have offered him a three-year deal worth €42 million (£27 million) in total.
Aubameyang has three years left on his deal with Dortmund, but the Bundesliga club "have refused to rule out" his sale.
ESPN FC's Jonathan Johnson supplied further details:
Jonathan Johnson @Jon_LeGossip
Interestingly, talks are thought to be advanced. Bild reckon transfer could be sealed soon after DFB Pokal final on May 27. #PEA #PSG #BVB2017-5-4 06:51:04
PSG are yet to replace Zlatan Ibrahimovic up front, and while Edinson Cavani has largely done a fine job leading the line on his own, another striker would take the pressure off the Uruguayan and restore some much-needed firepower.
Indeed, had Ibrahimovic stayed in the French capital, they may well have romped to another title, whereas they now face the prospect of losing out to AS Monaco.
Aubameyang has racked up 35 goals in all competitions this season, the majority of which have come in just 29 Bundesliga matches, per Squawka Football:
Squawka Football @Squawka
Pierre-Emerick Aubameyang has now scored 27 goals in the #Bundesliga this season; more than any other player. So consistent. 🔥 https://t.co/kN5KwpaRlu2017-4-22 17:50:23
The Gabon international's potent combination of pace, intelligent movement and clinical finishing makes him one of the most lethal frontmen in European football.
His capture would also help PSG advance their ambition of success in the UEFA Champions League—to have a realistic chance of winning the competition, you need players of his level.
Bleacher Report UK's Lars Pollmann has doubts over whether Aubameyang and Cavani could play together, but fellow German football expert Clark Whitney believes the pair could dovetail well:
Clark Whitney @Mr_Bundesliga
@LarsPollmann PEA played wide earlier in his career; no reason to believe he wouldn't be the one in the dual role with Cavani more fixed as Ibra once was2017-5-3 21:00:30
Clark Whitney @Mr_Bundesliga
@LarsPollmann If you can get 2 strikers who can play together because their qualities are different, but who can each play as a lone 9, it's brilliant2017-5-3 21:29:26
What's more, against the slightly weaker defences in Ligue 1 compared with the Bundesliga, there's no telling what kind of havoc Aubameyang could wreak.
Almost every elite club in Europe would most likely jump at the chance to sign Aubameyang this summer, but the Parisiens appear to have stolen a march on their rivals.
If PSG can get the deal sewn up early, they will have an excellent head start going into the transfer window and in all likelihood soften the blow of losing the Ligue 1 title to Monaco at the same time.
The Italian's agent, Donato Di Campli, rejected reports he had met with the Bianconeri but was seen in the vicinity of Juventus' hotel on Tuesday ahead of their Champions League clash with Monaco. Verratti is said to be at the top of Juve's wishlist this summer.
Sami Khedira and Miralem Pjanic have done an excellent job in midfield for the Old Lady this season, but Verratti could offer them a deep-lying playmaker in the vein of Andrea Pirlo.
He would be a brilliant signing, but PSG will be reluctant to part with him, and if they are willing to, then Juventus could face stiff competition from Barca and Bayern.